Question

If \(3^{-x}=2,\) what does \(3^{2 x}\) equal?

Short answer

3^{-x} = 2 \rightarrow 3^x = \frac{1}{2} \rightarrow 3^{2x} = \frac{1}{4}.

Step-by-step solution

- Understand the Given Equation

The given equation is \(3^{-x} = 2\). The objective is to find the value of \(3^{2x}\).

- Rewrite Using Properties of Exponents

Consider the exponent property \(a^{-b} = \frac{1}{a^b}\). Rewrite the equation \(3^{-x} = 2\) as \(\frac{1}{3^x} = 2\).

- Solve for \(3^x\)

To solve for \(3^x\), take the reciprocal of both sides: \(3^x = \frac{1}{2}\).

- Square Both Sides

To find \(3^{2x}\), square both sides of the equation \(3^x = \frac{1}{2}\). Thus, \(3^{2x} = \left( \frac{1}{2} \right)^2 = \frac{1}{4}\).

Key concepts

Properties of Exponents

Exponents are a way to express repeated multiplication of a number by itself. Knowing some key properties can help you manipulate and solve equations effectively. Here are some important properties:

• \textbf{Product of Powers:} When multiplying like bases, add their exponents. For example, \(a^m \times a^n = a^{m+n}\).
• \textbf{Quotient of Powers:} When dividing like bases, subtract the exponents. For example, \(a^m / a^n = a^{m-n}\).
• \textbf{Power of a Power:} When raising a power to another power, multiply the exponents. For example, \([a^m]^n = a^{m \times n}\).
• \textbf{Negative Exponents:} A negative exponent represents the reciprocal of the base raised to the positive exponent. Formally, \(a^{-n} = \frac{1}{a^n}\).

Understanding these properties will help simplify and solve exponential equations faster and easier.

Solving Exponential Equations

Solving exponential equations often involves expressing the equation in terms of the same base or using logarithms. For instance, in the given problem, you had the equation \(3^{-x} = 2\). The key to solving such equations is understanding and applying the properties of exponents and sometimes taking reciprocals. Let’s break this down:

1. The given equation \(3^{-x} = 2\) can be rewritten using the negative exponent property: \(3^{-x} = \frac{1}{3^x} = 2\).
2. To isolate \(3^x\), take the reciprocal of both sides: \(3^x = \frac{1}{2}\).
3. The goal here is to find the value of \(3^{2x}\). To achieve this, we square both sides of the equation \(3^x = \frac{1}{2}\), resulting in \(3^{2x} = \frac{1}{4}\).

Through these steps, exponential equations can be simplified and solved by carefully manipulating the bases and exponents.

Reciprocals in Exponents

Reciprocals are very helpful when dealing with negative exponents or when you need to simplify certain expressions. A reciprocal of a number \(a\) is \(1/a\). This concept is particularly useful in exponential equations where a negative exponent can be easier handled by expressing it as a reciprocal.

In our example, the expression \(3^{-x} = 2\) was rewritten using the reciprocal property: \(3^{-x} = \frac{1}{3^x} = 2\). Next, taking the reciprocal of both sides allows us to solve for \(3^x\):

• Reciprocal of \(3^x\) is \( \frac{1}{3^x} = 2\).
• Transform it to be \(3^x = \frac{1}{2}\).
• To take it to the final goal of finding \(3^{2x}\), square both sides of the equation.

This results in \(3^{2x} = ( \frac{1}{2})^2 = \frac{1}{4}\). Understanding how and when to switch to reciprocals in exponents makes it easier to simplify and solve these types of equations.